# Tagged Questions

For requesting, clarifying, and comparing definitions of mathematical terms.

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### Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I've been trying to work out what generic means but I'm not making much progress. You can find an example of the usage of the word generic for example here: "School on Generic Singularities in ...
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### Can limits be defined in a more algebraic way, instead of using the completely analytic $\delta$-$\epsilon$ definition?

Let $(X,d_X), (Y,d_Y)$ be metric spaces. Let $f:E\subseteq X\to Y$ and $a\in X$. We say that $\lim_{x\to a}f(x)=L$ if and only if for every $\epsilon >0$ there exists a $\delta>0$ such that ...
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### Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
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### Nothing on the web; What is a Ruffini Radical

Surprisingly, it's not clearly defined online. The first thing that comes up is Abel-Ruffini theorem, which only refers to "radicals" and not RUFFINI radicals. Ian Stewart's book has it appear out of ...
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### finite presentations of algebraic structures

I've read an algebraic variety $A$ is finitely presented if there's a coequalizer diagram $$F(m)\rightrightarrows F(n)\twoheadrightarrow A$$ where $F(n)$ is the free model on $n$ generators. I'm ...
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### Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
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### Does the metric in the Theory of Relativity actually satisfy the definition of a metric?

Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ...
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### Quick clarification on the definition of vector field

I am having a class on differential geometry and another on ODEs In the ODE class, if we were given something of the type $$\dot x = x^2$$ The professor refers to $x^2$ as the vector field. In ...
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### Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...
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### Why do we need $\sigma$-algebra in the definition of a measurable space?

A measurable space is a triple $(M, \Sigma, \mu)$. I am confused by one thing, to my knowledge almost everything you think of is measurable. So why do we need $\Sigma$? Why is that a measurable set ...
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### What is the group $O(2)$?

I have learnt about the group $$SO(2) = \left\{\pmatrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)} : \theta \in \mathbb{R} \right\}.$$ This group has do with rotations. ...
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### What is “analytic vector for closed operator”?

I need the defenition of "analytic vector of closed operator that acts on Hilbert space". I cant find it in google and in my textbooks (Khelemsky "Lectures And Exercises on Functional Analysis"), I ...
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### Matching the definition of hom-functor with how these are used when defining adjuncts

I have a problem matching the definition of a hom-functor (from nlab) with how this concept it used in the definition of adjunction (from nlab): The hom-functor is defined on $C^{\text{op}}\times C$, ...
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### Defining adjoint functors: What does “natural bijection” mean?

Take the following definition of adjunction from the nlab This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for ...
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### Is $f(x) = 2x+1$ injective? Is it surjective? [closed]

How would I answer this? I know what it means to be surjective and injective. Is the function $f(x)=2x+1$ injective? Is it surjective? Give reasons for your answers. I assume they are both because ...
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### Meaning of denseness in a $L^p$ spaces?

I am currently studyind Density theorems in $L^p$ - spaces. In that, I have encountered a theorem which goes like this - The space of integrable simple functions is dense in $L^p$(E, $\mathcal{A}$ ,...
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